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Municipal bonds (munis) issued by state and local governments in the U.S. are an important sector in fixed-income markets because the interest income is exempt from federal taxation. Issuers are not allowed to “arbitrage” the market – that is, they are not allowed to issue debt at low yields due to the exemption and turn around and invest the funds at a higher rate. The tax- exempt status of munis appeals to wealthy individuals and some institutions as part of their tax management strategies. Investors might also be exempt from state and local income taxes if they reside in the issuer's locality. That can definitely matter to those of you living in certain high-tax East Coast cities and states.

Investors often evaluate a municipal bond based on its equivalent taxable yield (ETY) statistic. The intent is to be able to compare directly the yield on a tax-exempt bond to otherwise comparable fully taxable corporate offerings. This comparison is not as easy as it might sound because the bond rating agencies have different criteria for each bond type – that is, a double A-rated muni does not necessarily have the same projected probability of default (and recovery rate) as a double A-rated corporate bond. Nevertheless, let's see how to calculate an ETY both in practice and in theory.

Assume now that the 4% and 1%, annual payment, 4-year bonds priced at 99.342 and 88.499, respectively, really are tax-exempt munis and not fully taxable corporate bonds. The commonly used ETY statistic, which we can call the street version because it is widely used in practice, is the after-tax yield divided by one minus the assumed ordinary income tax rate. Note that the after-tax yield on the muni is not just its quoted, or pretax, yield to maturity. When the muni is purchased at a premium or discount, there still are federal tax implications. Assume the investor holds the bond to maturity (selling prior to maturity can also generate taxable capital gains and losses), the ordinary tax rate is 25%, and the capital gains rate is 15%.

The after-tax yield to maturity on the 4% muni is 4.159%, the solution for aty.

Each interest payment of 4 (percent of par value) is now exempt from federal taxes. The small gain from buying at 99.342 and redeeming at 100 is taxed at the capital gains rate because the de minimis OID rule applies. The street ETY for this bond is 5.545%.

If the 1% muni bond priced at 88.499 is newly issued, it would be classified as OID. Then the movement along the constant-yield trajectory would be reported each year as tax-exempt interest income. Assume that this muni instead is a seasoned offering originally issued at par value (or de minimis OID) and so it is now purchased at a market discount. Its after-tax yield to maturity is 3.444%, the internal rate of return on the after-tax cash flows.

Notice that the market discount of 11.501 is taxed at the assumed ordinary income rate. This reflects tax rules since 1993. If this muni had been purchased before 1993, the market discount on the held-to-maturity bond would be taxed at the capital gains rate. If purchased after that date, it is taxed as ordinary income.

If this tax treatment seems odd to you, I agree completely. I'm really disappointed with the federal government because these tax rules do not make economic sense. In principle, the “gain” from buying at a market discount is just deferred interest income and should be tax exempt in my opinion. Suppose these two bonds were issued by the same state government – they mature on the same date and entail the same credit risk. Their after-tax yields to maturity should be very similar (perhaps differing due to liquidity). But the 1 % market discount bond is penalized significantly because the “gain” is taxed as ordinary income – not as tax-exempt interest income. Its street ETY is only 4.592%.

Let's now return to the intellectual safety of theoretical bond math. There

is another way to think about calculating the ETY on a tax-exempt bond. This is to solve for the internal rate of return on a taxable offering that generates the same after-tax cash flows as the muni. Let's do this first for the 4% tax-exempt bond priced at 99.432. We need to solve this equation for ETY:

The pretax coupon payments would have to be 5.333 [= 4/(1 – 0.25)] to equal 4 on a tax-exempt basis. The redemption amount of 100 does not need to be adjusted because both the taxable corporate and the “tax-exempt” muni face capital gains taxation on the de minimis OID. This ETY turns out to be 5.521%, just a bit lower than the street ETY of 5.545%.

The difference between the commonly used street ETY and what I suggest to be a better version becomes more relevant when we work through the deeper discount 1% muni bond priced at 88.499.

The annual coupon payment on the taxable corporate bond would be 1.333 [= 1/(1 – 0.25)] to give the same after-tax cash flow as the 1% muni bond. Again the principal does not need adjustment because the market discount on both bonds is taxed at the ordinary income rate. This “better” ETY is 4.542%, 5 basis points lower than the street ETY of 4.592%.

Why is this version of ETY better than the street ETY, which is commonly used in practice? First, it is the more natural formulation of the bond math problem in my opinion. But more important, it allows the analyst to include a term structure of tax rates. Suppose that there is a scheduled increase in ordinary income tax rates for the wealthy investor, going up from 25% to 35% in the third year and 40% in the fourth. The “more theoretically correct” ETY on the 4% muni becomes 6.027%, the solution for ETY in this expression.

It's not obvious how to handle that assumption in the street ETY calculation (divide by some weighted average of the tax rates?). I suppose a street ETY is relevant information, and need not be relegated to “data” status, but better bond math should give us better information leading to better decisions.


I recall being told many years ago that everything interesting in finance boils down to either options or taxes. Like most overstatements, there is no doubt some truth there. But in my experience, bond taxation is often given short shrift in investments and fixed-income textbooks. Perhaps an indication of this reluctance to deal carefully with taxation is that the widely used Bloomberg Yield and Spread Analysis page reports misleading results for the projected after-tax rates of return on some market discount bonds, at least for U.S. investors.

Now on to yield curve analysis, where we predictably neglect tax effects.

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